In physics, an operator is almost always either a square matrix or a linear mapping from one space of functions (often on $\mathbb{R}^N$ or $\mathbb{C}^N$) to the same or other like space of functions. Operators serve as *observables* and as *time evolution operators* in Quantum Mechanics. This tag ...

learn more… | top users | synonyms

35
votes
2answers
783 views

Physical interpretation of different selfadjoint extensions

Given a symmetric (densely defined) operator in a Hilbert space, there might be quite a lot of selfadjoint extensions to it. This might be the case for a Schrödinger operator with a "bad" potential. ...
32
votes
5answers
3k views

Is mass an observable in Quantum Mechanics?

One of the postulates of QM mechanics is that any observable is described mathematically by a hermitian linear operator. I suppose that an observable means a quantity that can be measured. The mass ...
31
votes
3answers
2k views

A “Hermitian” operator with imaginary eigenvalues

Let $${\bf H}=\hat{x}^3\hat{p}+\hat{p}\hat{x}^3$$ where $\hat{p}=-id/dx$. Clearly ${\bf H}^{\dagger}={\bf H}$, because ${\bf H}={\bf T} + {\bf T}^{\dagger}$, where ${\bf T}=\hat{x}^3\hat{p}$. In this ...
23
votes
4answers
2k views

Density matrix formalism

The density matrix $\hat{\rho}$ is often introduced in textbooks as a mathematical convenience that allows us to describe quantum systems in which there is some level of missing information. ...
23
votes
3answers
1k views

Why does non-commutativity in quantum mechanics require us to use Hilbert spaces?

I am reading Why we do quantum mechanics on Hilbert spaces by Armin Scrinzi. He says on page 13: What is new in quantum mechanics is non-commutativity. For handling this, the Hilbert space ...
20
votes
6answers
2k views

Why are Only Real Things Measurable?

Why can't we measure imaginary numbers? I mean, we can take the projection of a complex wave to be the "viewable" part, so why are imaginary numbers given this immeasurable descriptor? Namely with ...
20
votes
2answers
4k views

Time as a Hermitian operator in QM?

In non-relativistic QM, on one hand we have the following relations: $$\langle x | P | \psi \rangle ~=~ -i \hbar \frac{\partial}{\partial x} \psi(x),$$ $$\langle p | X | \psi \rangle ~=~ i \hbar ...
19
votes
7answers
26k views

What is the Physical Meaning of Commutation of Two Operators?

I understand the mathematics of commutation relations and anti-commutation relations, but what does it physically mean for an observable (self-adjoint operator) to commute with another observable ...
19
votes
4answers
2k views

Applications of the Spectral Theorem to Quantum Mechanics

I'm currently learning some basic functional analysis. Yesterday I arrived at the spectral theorem of self-adjoint operators. I've heard that this theorem has lots of applications in Quantum ...
18
votes
2answers
2k views

Intuitive meaning of Hilbert Space formalism

I am totally confused about the Hilbert Space formalism of Quantum Mechanics. Can somebody please elaborate on the following points: The observables are given by self-adjoint operators on the ...
18
votes
1answer
1k views

Is the Uncertainty Principle valid for information about the past?

My layman understanding of the Uncertainty Principle is that you can't determine the both the position and momentum of a particle at the same point in time, because measuring one variable changes the ...
17
votes
4answers
4k views

How to tackle 'dot' product for spin matrices

I read a textbook today on quantum mechanics regarding the Pauli spin matrices for two particles, it gives the Hamiltonian as $$ H = \alpha[\sigma_z^1 + \sigma_z^2] + ...
17
votes
3answers
3k views

Equivalence of canonical quantization and path integral quantization

Consider the real scalar field $\phi(x,t)$ on 1+1 dimensional space-time with some action, for instance $$ S[\phi] = \frac{1}{4\pi\nu} \int dx\,dt\, (v(\partial_x \phi)^2 - \partial_x\phi\partial_t ...
15
votes
1answer
2k views

Rigged Hilbert space and QM

Are there any comprehensive texts that discuss QM using the notion of rigged Hilbert spaces? It would be nice if there were a text that went through the standard QM examples using this structure.
15
votes
2answers
581 views

The formal solution of the Schrodinger equation

Consider the Schrödinger equation (or some equation in Schrödinger form) written down as $$ \tag 1 i \partial_{0} \Psi ~=~ \hat{H} \Psi . $$ Usually, one likes to write that it has a formal solution ...
15
votes
3answers
5k views

Why do we use operators in quantum mechanics?

In classical mechanics, physical quantities, such as, e.g. the coordinates of position, velocity, momentum, energy, etc, are real numbers, but in quantum mechanics they become operators. Why is this ...
14
votes
1answer
1k views

Self-adjoint and unbounded operators in QM

An operator $A$ is said to be self-adjoint if $(\chi,A\psi)=(A\chi,\psi)$ for $\psi, \chi \in D_A$ and $D_A=D_{A^\dagger}$. But for the free particle momentum operator $\hat{p}$ these inner products ...
14
votes
2answers
2k views

Does every hermitian operator represent a measurable quantity?

In Quantum mechanics, observables are represented by hermitian operator. But does every hermitian operator represent a observable? If not , how do we know that whether a hermitian operator represent ...
14
votes
1answer
1k views

Discreteness of set of energy eigenvalues

Given some potential $V$, we have the eigenvalue problem $$ -\frac{\hbar^2}{2m}\Delta \psi + V\psi = E\psi $$ with the boundary condition $$ \lim_{|x|\rightarrow \infty} \psi(x) = 0 $$ If we ...
13
votes
4answers
895 views

Can momentum have a complex expectation value?

I'm making examples of wave functions to incorporate in a QM exam. I came up with the following wave function, which gives me some troubles: $$\psi(x,0) = \begin{cases} A(a-x), & -a \leq x \leq ...
13
votes
4answers
991 views

Applying an operator to a function vs. a (ket) vector

I have a question regarding the effect of quantum mechanical operators. The definition that I'm familiar with says that an operator $A$ acts on a vector from a Hilbert space, $|\psi\rangle$, and the ...
13
votes
2answers
2k views

What does the Canonical Commutation Relation (CCR) tell me about the overlap between Position and Momentum bases?

I'm curious whether I can find the overlap $\langle q | p \rangle$ knowing only the following: $|q\rangle$ is an eigenvector of an operator $Q$ with eigenvalue $q$. $|p\rangle$ is an eigenvector of ...
13
votes
4answers
1k views

Really how can an observable quantity be equal to an operator?

A wave-function can be written as $$\Psi = Ae^{-i(Et - px)/\hbar}$$ where $E$ & $p$ are the energy & momentum of the particle. Now, differentiating $\Psi$ w.r.t. $x$ and $t$ respectively, ...
13
votes
4answers
1k views

Energy is actually the momentum in the direction of time?

By comparatively examining the operators a student concludes that `Energy is actually the momentum in the direction of time.' Is this student right? Could he be wrong?
13
votes
2answers
3k views

Meaning of the anti-commutator term in the uncertainty principle

What is the meaning, mathematical or physical, of the anti-commutator term? $\langle ( \Delta A )^{2} \rangle \langle ( \Delta B )^{2} \rangle \geq \dfrac{1}{4} \vert \langle [ A,B ] \rangle \vert^{2} ...
12
votes
3answers
5k views

Why do we use Hermitian operators in QM?

Position, momentum, energy and other observables yield real-valued measurements. The Hilbert-space formalism accounts for this physical fact by associating observables with Hermitian ('self-adjoint') ...
12
votes
3answers
2k views

How does one determine ladder operators systematically?

In textbooks, the ladder operators are always defined," and shown to 'raise' the state of a system, but they are never actually derived. Does one find them simply by trial and error? Or is there a ...
12
votes
5answers
12k views

How to get the position operator in the momentum representation from knowing the momentum operator in the position representation?

I know that $$\tag{1}\hat{p}~=~-i\hbar \frac{\partial}{\partial x}~.$$ How can I get $$\tag{2}\hat{x}~=~i\hbar \frac{\partial}{\partial p}~?$$ I think this simple and I'm just over thinking it, ...
12
votes
1answer
186 views

Explaining causal completion axiom in Haag-Kastler axioms?

There are several variants of the Haag-Kastler axioms for algebraic quantum field theory. Usually one associates an algebra $\mathcal{A}(O)$ to each open region $O$ of spacetime. An often-suggested ...
11
votes
2answers
3k views

Differences between symmetric, Hermitian, self-adjoint, and essentially self-adjoint operators

I am a physicist. I always heard physicists used the terminology "symmetric", "Hermitian", "self-adjoint", and "essentially self-adjoint" operators interchangeably. Actually what is the difference ...
11
votes
2answers
756 views

In what sense is the path integral an independent formulation of Quantum Mechanics/Field Theory?

We are all familiar with the version of Quantum Mechanics based on state space, operators, Schrodinger equation etc. This allows us to successfully compute relevant physical quantities such as ...
11
votes
1answer
8k views

Evolution operator for time-dependent Hamiltonian

When I studied QM I'm only working with time independent Hamiltonians. In this case the unitary evolution operator has the form $$\hat{U}=e^{-\frac{i}{\hbar}Ht}$$ that follows from this equation $$ ...
11
votes
2answers
742 views

How are anyons possible?

If $|ψ\rangle$ is the state of a system of two indistinguishable particles, then we have an exchange operator $P$ which switches the states of the two particles. Since the two particles are ...
11
votes
1answer
620 views

What is the value of a quantum field?

As far as I'm aware (please correct me if I'm wrong) quantum fields are simply operators, constructed from a linear combination of creation and annihilation operators, which are defined at every point ...
11
votes
2answers
491 views

Eigenstate of field operator in QFT

Why don't people discuss the eigenstate of the field operator? For example, the real scalar field the field operator is Hermitian, so its eigenstate is an observable quantity.
11
votes
1answer
335 views

Mathematical motivation of OPE?

In Peskin & Schroeder (and also Cheng which I have skimmed through) they motivate the Operator Product Expansion with a lot of words. Is there any way to motivate it mathematically, e.g. Taylor ...
11
votes
1answer
122 views

Metric interpretation of self-adjoint extensions?

I am wondering if beyond physical interpretation, the one dimensional contact interactions (self-adjoint extensions of the the free Hamiltonian when defined everywhere except at the origin) have a ...
11
votes
1answer
319 views

Shape of the state space under different tensor products

I am currently studying generalized probabilistic theories. Let me roughly recall how such a theory looks like (you can skip this and go to "My question" if you are familiar with this). Recall: In a ...
10
votes
5answers
2k views

What exactly is $\hat{\psi}^\dagger(x)$? An operator or a function?

I've recently read Cohen-Tannoudji on quantum mechanics to try to better understand Dirac notation. A homework problem is giving me some trouble though. I'm unsure if I've learned enough yet to ...
10
votes
2answers
437 views

Is there any theorem that suggests that QM+SR has to be an operator theory?

UPDATE To make my question more precise, I'll define what I mean by an operator theory: An operator theory is a theory in which the dynamical objects are operators, i.e., the equations of motion ...
10
votes
1answer
995 views

Operator Ordering Ambiguities

I have been told that $$[\hat x^2,\hat p^2]=2i\hbar (\hat x\hat p+\hat p\hat x)$$ illustrates operator ordering ambiguity. What does that mean? I tried googling but to no avail.
10
votes
2answers
631 views

Non-separable solutions of the Schroedinger equation

I'm studying an undergraduate Quantum Mechanics course and I have some doubts about the solution of the Schroedinger equation by the separation of variables method. If we suppose that the solutions ...
10
votes
4answers
802 views

Why are the inner products of the eigenfunctions of an operator with a discrete eigenvalue spectrum guaranteed to exist?

I was reading through a textbook, and the statement was made that the inner products are guaranteed to exist if the eigenvalue spectrum of the operator is discrete. I have come across no support for ...
10
votes
1answer
1k views

“An operator is hermitian”. Implications?

Alastair Rae states that there are 4 postulates of Quantum Mechanics in his text on the subject matter. The first part of his second postulate can be stated as: Every dynamical variable may be ...
9
votes
4answers
5k views

Difficulties with bra-ket notation

I have started to study quantum mechanics. I know linear algebra,functional analysis, calculus, and so on, but at this moment I have a problem in Dirac bra-ket formalism. Namely, I have problem with ...
9
votes
4answers
446 views

Where does the $i$ come from in the Schrödinger equation?

I am currently trying to follow Leonard Susskind's "Theoretical Minimum" lecture series on quantum mechanics. (I know a bit of linear algebra and calculus, so far it seems definitely enough to follow ...
9
votes
2answers
362 views

Motivation to introduce von Neumann algebras in addition to $C^*$algebras?

Observables are self-adjoint elements of a $C^*$algebra. As such, this structure seems sufficient to describe physics. A theorem by Gelfand and Naimark says that a $C^*$algebra can always be ...
9
votes
3answers
505 views

What is the physical meaning of weak expectation values?

In the two-state formalism of Yakir Aharonov, the weak expectation value of an operator $A$ is $\frac{\langle \chi | A | \psi \rangle}{\langle \chi | \psi \rangle}$. This can have bizarre properties. ...
9
votes
2answers
635 views

Regularisation of infinite-dimensional determinants

Can a regularisation of the determinant be used to find the eigenvalues of the Hamiltonian in the normal infinite dimensional setting of QM? Edit: I failed to make myself clear. In finite ...
9
votes
0answers
326 views

How can I write a Gaussian state as a squeezed, displaced thermal state

I would like to write a Gaussian state with density matrix $\rho$ (single mode) as a squeezed, displaced thermal state: \begin{gather} \rho = \hat{S}(\zeta) \hat{D}(\alpha) \rho_{\bar{n}} ...